Proof of Well-Posedness of the Allen-Cahn Hamilton-Jacobi Equation on the Torus

Each result cited is universally quantified over the data in its own statement and is applied here to the data at hand; the notation is that of the statement.

Claim 1.

(a) The nonlinearity. The Cube Map is a Monotone Nonlinearity on the Sobolev Hilbert Triple of the Torus is formed for a dimension $n$ with $1\le n$ and $n\le3$, for the Hilbert triple of The Square-Integrable and Sobolev Spaces of the Torus Form a Hilbert Triple (i), and for a real number $b$ with $0\le b$; all three conditions hold for the present data, and the map called $B$ there is, by The Cube Map is a Monotone Nonlinearity on the Sobolev Hilbert Triple of the Torus (i), the map $B$ of the present statement. Hence The Cube Map is a Monotone Nonlinearity on the Sobolev Hilbert Triple of the Torus (v) gives that $B$ is a monotone nonlinearity for $(H,V,A)$.

(b) The Lipschitz map. The number $|\kappa|$ is nonnegative by claim 1 of Properties of the Absolute Value in an Ordered Field. Let $X,Y\in L^{2}(\mathbb{T}^{n})$. The space $L^{2}(\mathbb{T}^{n})$ is a real vector space by The Riesz–Fischer Theorem (i), and scalar multiplication distributes over differences of vectors by the vector space axioms together with Elementary Identities in a Vector Space, so

$$L(X)-L(Y)=(-\kappa)X-(-\kappa)Y=(-\kappa)(X-Y).$$

Hence, by Real Inner Product Spaces (iii) and claim 4 of Elementary Identities in a Real Inner Product Space,

$$d_{L^{2}}\bigl(L(X),L(Y)\bigr)=\lVert(-\kappa)(X-Y)\rVert_{L^{2}}=|-\kappa|\;\lVert X-Y\rVert_{L^{2}}=|\kappa|\;d_{L^{2}}(X,Y),$$

the last equality by claim 2 of Properties of the Absolute Value in an Ordered Field and Real Inner Product Spaces (iii) again. Thus $L$ is Lipschitz with constant $|\kappa|$ from $(L^{2}(\mathbb{T}^{n}),d_{L^{2}})$ to itself.

(c) The hypotheses of the general theorem. Put $\ell=|\kappa|$. Comparing with the hypotheses of Well-Posedness for a Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple: the Hilbert triple $(H,V,A)$ is the one fixed in the present statement and satisfies the standing hypothesis Hilbert Triples: Standing Notation and Background (ii) by The Square-Integrable and Sobolev Spaces of the Torus Form a Hilbert Triple (i); the reals $\lambda_{0}$ and $C_{g}$ satisfy $0<\lambda_{0}$ and $0\le C_{g}$; $\omega_{g}$ is a modulus of continuity and $g$ is a function on $V=H^{1}(\mathbb{T}^{n})$ with $|g(X)|\le C_{g}$ and $|g(X)-g(Y)|\le\omega_{g}(|X-Y|_{V})$, since $|\cdot|_{V}$ is $\lVert\cdot\rVert_{H^{1}}$; the map $B:V\to H$ is a monotone nonlinearity by (a); $\ell$ is nonnegative and $L:H\to H$ is Lipschitz with constant $\ell$ by (b); and $\mathrm{Sym}(V)$ is $\mathrm{Sym}(H^{1}(\mathbb{T}^{n}))$. The function $F$ of the present statement is therefore precisely the function $F$ of that theorem formed for these data; it is defined, and is a degenerate elliptic second-order equation operator on $H$ relative to $(H,V,A)$, by the reference made there to A Hamilton-Jacobi Operator with a Monotone Nonlinearity and a Lipschitz Perturbation Satisfies the Comparison Hypotheses (i). Every hypothesis of that theorem is thus satisfied by the present data, which is claim 1.

Claim 2. By (c), claim 1 of Well-Posedness for a Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple applies to the present data and provides a function $u$ on $H=L^{2}(\mathbb{T}^{n})$ that is a viscosity solution of $F$ on $L^{2}(\mathbb{T}^{n})$, satisfies $|u(X)|\le C$ for every $X\in L^{2}(\mathbb{T}^{n})$, where $C=\tfrac{C_{g}}{\lambda_{0}}$ is the quotient formed in that theorem and is nonnegative there, and is uniformly continuous on $L^{2}(\mathbb{T}^{n})$ with respect to $d_{H}$, which is $d_{L^{2}}$, and the metric of Real Hilbert Spaces: Standing Notation and Background (i). Claim 3 of the same theorem, applied to this $u$ and to a viscosity solution $u'$ of $F$ on $L^{2}(\mathbb{T}^{n})$ that is continuous on $L^{2}(\mathbb{T}^{n})$ and for which some $C''\in\mathbb{R}$ satisfies $|u'(X)|\le C''$ for every $X\in L^{2}(\mathbb{T}^{n})$, gives $u'(X)=u(X)$ for every $X\in L^{2}(\mathbb{T}^{n})$. This is claim 2.

Claim 3. Let $v\in C^{2}_{\mathrm{per}}$ and put $X=[v|_{Q}]$.

(a) The Square-Integrable and Sobolev Spaces of the Torus Form a Hilbert Triple (ii), applied to $v$, gives $v\in C_{\mathrm{per}}$, $\Delta v\in C_{\mathrm{per}}$, $v-\Delta v\in C_{\mathrm{per}}$ with $(v-\Delta v)|_{Q}\in\mathcal{L}^{2}(\mathbb{T}^{n})$, and $X\in H^{1}(\mathbb{T}^{n})$ with

$$X\in D(A),\qquad AX=\bigl[(v-\Delta v)|_{Q}\bigr].$$

In particular $v|_{Q}\in\mathcal{L}^{2}(\mathbb{T}^{n})$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i), so $v|_{Q}$ is a representative of $X$.

(b) By Elementary Properties of Lattice-Periodic Functions (i), applied to pairs of members of $C_{\mathrm{per}}$, the pointwise products $v^{2}=vv$ and $v^{3}=v^{2}v$ lie in $C_{\mathrm{per}}$, and then so do the real multiples $b\,v^{3}$, $-(\kappa-1)v$ and $-\Delta v$ and their sum $-\Delta v+b\,v^{3}-(\kappa-1)v$. Its restriction to $Q$ therefore lies in $\mathcal{L}^{2}(\mathbb{T}^{n})$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i).

(c) Since $v|_{Q}$ is a representative of $X$ and $(v|_{Q})^{3}=(v^{3})|_{Q}$ valuewise, The Cube Map is a Monotone Nonlinearity on the Sobolev Hilbert Triple of the Torus (i) gives $B(X)=b\,[(v^{3})|_{Q}]$; and $L(X)=-\kappa\,[v|_{Q}]$ by the statement.

(d) The class map of The Lebesgue Space Lᵖ(μ) (ii) is additive and homogeneous, $L^{2}(\mathbb{T}^{n})$ carrying the vector space structure of The Riesz–Fischer Theorem (i); and restriction to $Q$ commutes valuewise with pointwise sums and real multiples. Hence, by (a) and (c),

$$AX+B(X)+L(X)=\Bigl[\bigl((v-\Delta v)+b\,v^{3}-\kappa\,v\bigr)\big|_{Q}\Bigr].$$

For every $y\in\mathbb{R}^{n}$, distributivity and the field axioms of The Real Numbers: Standing Notation and Background (i) give $v(y)-\kappa\,v(y)=(1-\kappa)v(y)=-(\kappa-1)v(y)$, whence

$$\bigl(v(y)-\Delta v(y)\bigr)+b\,(v(y))^{3}-\kappa\,v(y)=-\Delta v(y)+b\,(v(y))^{3}-(\kappa-1)v(y).$$

The two maps being equal valuewise, so are their restrictions to $Q$ and hence their classes, and therefore

$$AX+B(X)+L(X)=\Bigl[\bigl(-\Delta v+b\,v^{3}-(\kappa-1)\,v\bigr)\big|_{Q}\Bigr],$$

which together with (a) and (b) is claim 3.